Question: Simplify the following expression and state the condition under which the simplification is valid: $z = \dfrac{a^2 - 5a - 6}{a^2 - 4a - 5}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{a^2 - 5a - 6}{a^2 - 4a - 5} = \dfrac{(a - 6)(a + 1)}{(a - 5)(a + 1)} $ Notice that the term $(a + 1)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(a + 1)$ gives: $z = \dfrac{a - 6}{a - 5}$ Since we divided by $(a + 1)$, $a \neq -1$. $z = \dfrac{a - 6}{a - 5}; \space a \neq -1$